Embeddability and Stresses of Graphs

Gluck (1975) has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. We show that linklessly embeddable graphs are generically 4-stress free. Both of these results are corollaries of the following theorem: every K_{r+2}-minor free graph is generically r-stress free for 0<r<5. (This assertion is false for r>5.) We give an equivalent formulation of this theorem in the language of symmetric algebraic shifting and show that its analogue for exterior algebraic shifting also holds. Some further extensions are detailed.